Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

Choose an integer number between 0 and 100. You win a prize if your number is equal or closest to 2/3 of the average number chosen by all other participants. What number should you choose?

If you think that the other participants will choose a random number within the range, the average will be 50. Hence you choose 33. That seems right, intuitively, to many people. But hang on. Just as you chose 33, so presumably will other participants, at least on average, based on your same line of reasoning. So if the average number chosen by all participants is 33, then the smart thing to do is to choose 22.

But do you really think you are smarter than the others? Just as you figured out that 22 is the smart choice, so will others, at least on average. So the super smart thing to do is to choose 15. But … We are heading towards 0 (you get there after 12 iterations). Zero is the only rational choice to make if you don’t think you are smarter than the other participants.

You start to get the strong feeling that if you choose 0 you are not going to win the prize. This is because, although you don’t think you are smarter than most, it is reasonable to assume that at least some of the players are not as smart or rational as you. For example, if 10 per cent of players are totally naïve and choose a random number – 50 on average – then the overall average will be 5 and the right answer will be 3. However, if the rest of the players share your thoughts and assumptions, they will also choose 3, thereby increasing the average to 8 and the right answer to 5. Then you answer 5, but so will the rest, thus increasing the right answer to 6.

The process converges to 8. Well, 8 is the right answer if 90 per cent of players are as smart as you are and 10 per cent are totally naïve. If 20 per cent are naïve, the process converges to 14; with 30 per cent it converges to 18, and so on. But then it may also be the case that the less rational players are not totally naïve (Level 0 rationality) but, for example, exhibit Level 1 rationality, where the average answer is 33. In this case, with 10 per cent Level 1 players the process converges to 5; with 20 per cent to 9; with 30 per cent to 12, and so on. Of course, there are plenty more combinations, with varying proportions of players at Level 0, Level 1, Level 2 and so on. The higher the winning number, the larger is the percentage of less rational players in the game.

In an experiment conducted with Financial Times readers by economist Richard Thaler, made up of 1,476 participants, the winning number was in fact 13. This is roughly consistent with:

1. All players exhibit Level 3 rationality

OR 2. 80% are fully rational and 20% are totally naïve.

OR 3. 70% are fully rational and 30% exhibit Level 1 rationality.

Etc.

John Maynard Keynes, in Chapter 12 of his ‘General Theory of Employment, Interest and Money’, frames the paradox in terms of the money markets, in a more prosaic way:

“Professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one’s judgment, are really the prettiest, not even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practice the fourth, fifth and higher degrees.”

In other words, it is those who are able to best out-guess the best guesses of the rest of the crowd, who stand to win the prize. Or put another way, the ten pound note you spot lying on the floor might well be real after all. Nobody has picked it up yet because they have all assumed that someone else would have picked it up if it were real. You realise that everyone else is thinking like this, and you win yourself a tenner. Let’s call that super-rationality.

Exercise

Choose an integer number between 0 and 100. You win a prize if your number is equal or closest to 2/3 of the average number chosen by all other participants. What number should you choose?